Please note special day, time and location.   On a Kähler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. However, in the non-Kähler setting, such a link is not so obvious. I will discuss the existence of non-Kähler Hermitian metrics for which a certain proportionality relationship between the Chern and Riemannian scalar curvatures holds. The study of such metrics, in turn, leads to a general question concerning the behavior of the lowest eigenvalue of Schrödinger operators on compact Riemannian manifolds. This is joint work with Mike Dabkowski.